NUMERICAL DISCRETIZATION
AND STABILITY
By Kevin Brockman &
Dan Nesser
WHAT IS NUMERICAL
DISCRETIZATION?
•  Discretization concerns the process of transferring
continuous models and equations into discrete
counterparts.
DISCRETIZATION IN
WEATHER
•  Forecast Grids & Numerical Weather Prediction
INITIAL VALUE PROBLEM
•  Hyperbolic and Parabolic PDEs are initial value
problems
•  One in which the solution is obtained by using the
known initial values and advancing in time.
FINITE DIFFERENCE
METHOD
•  Two fundamental questions:
a) Is the FDE consistent with the PDE?
b) For any given time t>0, will the solution U of the FE
converge to u as change in x and change in t
approaches 0?
TRUNCATION ERROR &
CONSISTENCY
•  Local Truncation Error should go to zero
•  Can be verified with Taylor Series expansion
CONVERGENCE &
STABILITY CRITERIA
•  This is how we need to answer our second question.
•  Before we look at convergence, we need to answer
the question of computational stability.
COMPUTATIONAL
STABILITY
•  One example
•  Ujn+1=(1-µ)Ujn+µUj-1n
•  This is our PDE from earlier in the reading, which tries to
estimate the advection equation
•  µ= cΔt/Δx: Called Courant number
COURANT NUMBER
•  If 0 ≤ µ ≤ 1 the FDE is considered bounded
•  Estimations are made through interpolation
•  Our FDE stays close to the PDE
•  If the above case is not true, then the FDE is considered
unbounded
•  Estimations are made through extrapolation
•  The FDE diverges away from the PDE
COMPUTATIONAL
STABILITY
•  We can now define Computational Stability
•  An FDE is computationally stable if the solution of the FDE
at a fixed time t = nΔt remains bounded as Δt
approaches zero.
•  The condition of |µ| being less than one is called
Courant-Fredrichs-Lewy (CFL) condition
CONVERGENCE
•  Stability is a critical part in determining convergence
•  “The stability of the FDE is the necessary and sufficient
condition for convergence”